June 6, 2023

# What is the Quotient Rule?

Quotient rule in calculus is a method used to find the derivative of a function in the form of the ratio of two differentiable functions. It is defined by the ratio of the quantity of the denominator divided by the square of the denominator function less the quantity of the numerator divided by the derivative of the numerator function.

It also follows the definition of the limit of the derivative.

Quotient rule can only be applied in functions that are expressed as a ratio or in the form of numerator and denominator in a pie or natural numbers; this rule cannot be applied. This is also called the division rule of differentiation.

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What is the Formula for Calculating Quotient Rule?

dxdy =vdudx- udvdx/ v2

Dy/dx =  derivative of y

V = variable

U = variable

Du/dx = derivative of u

Dv/dx = derivative of v

It can also be expressed as the denominator (Derivative of the Numerator) – Numerator (Derivative of the Denominator) / Denominator square.

## Methods to Prove the Quotient Rule Formula

There are three methods to Prove the Quotient rule formula:-

• Using derivative and limit properties:- Based on the Quotient Rule, the derivative of a quotient is the denominator divided by the numerator.
• Using implicit differentiation:- The implicit differentiation procedure involves treating one of the variables (usually x and y) as a function of the other in an equation with two variables (usually x and y)
• Using chain rule:- In other words, the chain rule helps us distinguish ‘composite functions’. For example, sin(x2) is a composite function because it can be decomposed into f(g(x)) when f(x)=sin(x) and g(x)=x2.

## When can we Use the Quotient Rule?

You can look up the derivative of a function whose derivative is divided by another function by using the quotient rule. Or,

To obtain the derivative of one function divided by another, such as u / v, you need to utilize the quotient rule.

## Is the Product Rule and the Quotient Rule the Same?

A derivative of a function can be found by using the product rule. A derivative of a function would be the product of the two (or more) functions above.

When two differentiable functions are ratios, we can find the derivative of the function based on the quotient rule.

## Example

Example 1:- d/dx [7x + 4/x2 + 5]

Solution:

The formula for derivatives:-

= d/dx [f/g] = gf’ – fg’/ g

So, f = 7x + 4

g = x2 + 5

f’ = 7

g’ = 2x

= (x2 + 5) (7) – (7x + 4) (2x) / (x2 + 5)2

= 7x2 + 35 – 14x2 – 8x/ (x2 + 5)2

= 35 -8x -7x2/ (x2 + 5)2

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